Abstract Let R be a commutative Noetherian ring, I an ideal of R, M an R-module (not necessary I-torsion) and K a finitely generated R-module with SuppR (K) ⊆ V(I). It is shown that if M is I-ET H-cofinite (i.e. ExtiR (R/I, M ) is finitely generated, for all i ≥ 0) and dim M ≤ 1, then the R-module Extn R(M, K) is finitely generated, for all n ≥ 0. As a consequence it is shown that if M is I-ET H-cofinite and FD≤1 (or weakly Laskerian), then the R-module Extn R(M, K) is finitely generated, for all n ≥ 0 which removes I-torsion condition of M from [3, Corollary 3.11] and [20, Theorem 2.8]. As an application to local cohomology, let Φ be a system of ideals of R and I ∈ Φ, if dim M/aM ≤ 1 (e.g., dim R/a ≤ 1) for all a ∈ Φ, then the R-modules ExtjR(Hi Φ(M ), K) are finitely generated, for all i ≥ 0 and j ≥ 0. A similar result is true for local cohomology modules defined by a pair of ideals.
